Strong embedding conjecture for permutation polytopes

Let $P$ be a $d$-dimensional permutation polytope. A permutation polytope is the convex hull of permutation matrices associated with a permutation group, and two polytopes may be combinatorially equivalent or lattice equivalent.

Strong embedding conjecture. There exists a permutation group $G\leq S_{2d}$ such that $P(G)$ is combinatorially equivalent, or more strongly lattice equivalent, to $P$.

The bound $2d$ is suggested to be sharp by the example of the $d$-cube, while the paper notes that the existence of some bound follows from an earlier proposition. The claim is presented as open.